Answer
False.
There are no nonzero values of $a$ and $b$ such that$$(a+b)^4=a^4+b^4.$$
Work Step by Step
Please note that if at least one of $a$ or $b$ is zero, say $b=0$, then the identity holds true:
$$(a+0)^4=a^4=a^4+0^4.$$
However, for nonzero values of $a$ and $b$, the identity does not hold because the binomial expansion of $(a+b)^4$ has the terms $a^{4-r}b^r$, $r=1, 2, 3$, which are nonzero.
